Group #14530950 — subgroups

Subgroups — #14530950

Order 2176.

124 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 124 conjugacy classes of subgroups — too many to draw. Only its 49 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763210	C₂	1	1	yes	maximal, characteristic
2	Q₁₆×Dic₁₇	C₂	1	1	yes	maximal, characteristic
3	#11762636	C₂	1	1	yes	maximal, characteristic
4	#11762635	C₂	1	1	yes	maximal, characteristic
5	#11762627	C₂	1	1	yes	maximal, characteristic
6	#11762613	C₂	1	1	yes	maximal, characteristic
7	C₁₇×#358	C₂	1	1	yes	maximal, characteristic
8	C₂×Q₁₆×C₁₇	C₂²	1	1	yes	characteristic
9	Q₈×Dic₁₇	—	2	1	no	—
10	#10587576	—	2	1	no	—
11	C₂×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
12	C₁₇×#107	C₂²	1	1	yes	characteristic
13	#10587506	—	2	1	no	—
14	#10587489	C₂²	1	1	yes	characteristic
15	#10587489	C₂²	1	1	yes	characteristic
16	C₈×Dic₁₇	C₂²	1	1	yes	characteristic
17	C₂×#63214	C₂²	1	1	yes	characteristic
18	#10587479	—	2	1	no	—
19	C₂×Q₈×C₁₇	—	2	1	no	—
20	Q₁₆×C₁₇	—	2	1	no	—
21	Q₁₆×C₁₇	Q₈	1	2	yes	—
22	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
23	C₄⋊C₄×C₁₇	—	2	1	no	—
24	C₄⋊Dic₁₇	—	2	1	no	—
25	C₄⋊Dic₁₇	—	2	1	no	—
26	C₄⋊Dic₁₇	—	4	1	no	—
27	C₄⋊Dic₁₇	—	4	1	no	—
28	#63225	—	4	1	no	—
29	C₄×Dic₁₇	—	4	1	no	—
30	C₄×Dic₁₇	D₄	1	1	yes	characteristic
31	C₂×#3622	D₄	1	1	yes	characteristic
32	C₁₆×C₁₇	—	2	1	no	—
33	#63214	—	2	1	no	—
34	Q₈×C₁₇	—	2	2	no	—
35	Q₈×C₁₇	—	4	1	no	—
36	C₂×C₄×C₁₇	—	4	1	no	—
37	C₂×C₄×C₁₇	—	4	1	no	—
38	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
39	C₂×Dic₁₇	—	4	1	no	—
40	C₂×Dic₁₇	—	4	1	no	—
41	C₂×Dic₁₇	D₈	1	1	yes	characteristic
42	C₂×Dic₁₇	D₈	1	1	yes	characteristic
43	C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
44	C₈×C₁₇	#41	1	1	yes	characteristic
45	#3622	—	2	1	no	—
46	#2229	—	17	1	no	maximal, Sylow
47	C₂²×C₁₇	C₂×D₈	1	1	yes	characteristic
48	C₄×C₁₇	—	4	2	no	—
49	C₄×C₁₇	—	8	1	no	—
50	C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
51	C₄×C₁₇	#137	1	1	yes	characteristic
52	Dic₁₇	—	2	1	no	—
53	Dic₁₇	—	2	1	no	—
54	Dic₁₇	—	8	1	no	—
55	Dic₁₇	—	8	1	no	—
56	#499	—	17	1	no	—
57	C₄×Q₁₆	—	17	1	no	—
58	#367	—	17	1	no	—
59	#366	—	17	1	no	—
60	C₄⋊C₁₆	—	17	1	no	—
61	#358	—	17	1	no	—
62	#358	D₁₇	1	1	yes	characteristic
63	C₂×C₁₇	#474	1	1	yes	characteristic
64	C₂×C₁₇	#508	1	1	yes	characteristic
65	C₂×C₁₇	#510	1	1	yes	characteristic
66	C₂×Q₁₆	C₂×D₁₇	1	1	yes	characteristic
67	#125	—	34	1	no	—
68	C₄×Q₈	—	34	1	no	—
69	C₂×C₁₆	—	17	1	no	—
70	C₂×C₁₆	C₂×D₁₇	1	1	yes	characteristic
71	#107	—	17	1	no	—
72	#107	—	17	1	no	—
73	#107	C₂×D₁₇	1	1	yes	characteristic
74	#106	—	34	1	no	—
75	#103	—	34	1	no	—
76	C₄×C₈	—	17	1	no	—
77	C₁₇	#2229	1	1	yes	characteristic, complemented, Sylow
78	C₂×Q₈	—	2	1	no	—
79	Q₁₆	—	2	1	no	—
80	Q₁₆	Dic₃₄	1	2	yes	—
81	C₂×C₈	—	17	1	no	—
82	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
83	C₄⋊C₄	—	2	1	no	—
84	C₄⋊C₄	—	34	1	no	—
85	C₄⋊C₄	—	34	1	no	—
86	C₄⋊C₄	—	68	1	no	—
87	C₄⋊C₄	—	68	1	no	—
88	C₄⋊C₄	—	68	1	no	—
89	C₄²	—	17	1	no	—
90	C₄²	—	68	1	no	—
91	C₁₆	—	2	1	no	—
92	C₁₆	—	34	1	no	—
93	Q₈	—	2	2	no	—
94	Q₈	—	4	1	no	—
95	C₂×C₄	—	4	1	no	—
96	C₂×C₄	—	4	1	no	—
97	C₂×C₄	—	17	1	no	—
98	C₂×C₄	—	17	1	no	—
99	C₂×C₄	—	68	1	no	—
100	C₂×C₄	—	68	1	no	—
101	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
102	C₈	—	34	1	no	—
103	C₈	C₂×Dic₃₄	1	1	yes	characteristic
104	C₈	#63254	1	1	yes	characteristic
105	C₂²	D₈×D₁₇	1	1	yes	characteristic
106	C₄	—	4	2	no	—
107	C₄	—	8	1	no	—
108	C₄	—	34	1	no	—
109	C₄	—	34	1	no	—
110	C₄	—	136	1	no	—
111	C₄	—	136	1	no	—
112	C₄	#10587564	1	1	yes	characteristic
113	C₄	#10587601	1	1	yes	characteristic
114	C₂	#11763087	1	1	yes	characteristic
115	C₂	#11763242	1	1	yes	characteristic
116	C₂	#11763248	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11763210

C₂

yes

maximal, characteristic

Q₁₆×Dic₁₇

C₂

yes

maximal, characteristic

#11762636

C₂

yes

maximal, characteristic

#11762635

C₂

yes

maximal, characteristic

#11762627

C₂

yes

maximal, characteristic

#11762613

C₂

yes

maximal, characteristic

C₁₇×#358

C₂

yes

maximal, characteristic

C₂×Q₁₆×C₁₇

C₂²

yes

characteristic

Q₈×Dic₁₇

—

#10587576

—

C₂×C₁₆×C₁₇

C₂²

yes

characteristic

C₁₇×#107

C₂²

yes

characteristic

#10587506

—

#10587489

C₂²

yes

characteristic

#10587489

C₂²

yes

characteristic

C₈×Dic₁₇

C₂²

yes

characteristic

C₂×#63214

C₂²

yes

characteristic

#10587479

—

C₂×Q₈×C₁₇

—

Q₁₆×C₁₇

—

Q₁₆×C₁₇

Q₈

yes

—

C₂×C₈×C₁₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

#63225

—

C₄×Dic₁₇

—

C₄×Dic₁₇

D₄

yes

characteristic

C₂×#3622

D₄

yes

characteristic

C₁₆×C₁₇

—

#63214

—

Q₈×C₁₇

—

Q₈×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

D₈

yes

characteristic

C₂×Dic₁₇

D₈

yes

characteristic

C₈×C₁₇

C₂×Q₈

yes

C₈×C₁₇

yes

—

—

C₂²×C₁₇

C₂×D₈

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

C₂²⋊Q₈

yes

characteristic

C₄×C₁₇

#137

yes

characteristic

Dic₁₇

—

Dic₁₇

—

Dic₁₇

—

Dic₁₇

—

#499

—

C₄×Q₁₆

—

#367

—

#366

—

C₄⋊C₁₆

—

#358

—

#358

D₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×Q₁₆

C₂×D₁₇

yes

characteristic

#125

—

C₄×Q₈

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₁₇

yes

—

—

C₂×D₁₇

yes

characteristic

#106

—

#103

—

C₄×C₈

—

C₁₇

#2229

yes

characteristic, complemented, Sylow

C₂×Q₈

—

Q₁₆

—

Q₁₆

Dic₃₄

yes

—

C₂×C₈

—

C₂×C₈

C₂²×D₁₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₁₆

—

C₁₆

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

D₄×D₁₇

yes

characteristic

102

C₈

—

103

C₈

C₂×Dic₃₄

yes

characteristic

104

C₈

#63254

yes

characteristic

105

C₂²

D₈×D₁₇

yes

characteristic

106

C₄

—

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

—

136

—

111

C₄

—

136

—

112

C₄

#10587564

yes

characteristic

113

C₄

#10587601

yes

characteristic

114

C₂

#11763087

yes

characteristic

115

C₂

#11763242

yes

characteristic

116

C₂

#11763248

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₁₇
commutator, cosocle	C₂×C₈×C₁₇	C₂³
Frattini	C₂×C₈	C₂²×D₁₇
Fitting	C₁₇×#358	C₂
socle, prime radical	C₂²×C₁₇	C₂×D₈
hypercenter	#358	D₁₇
hypocenter	C₁₇	#2229
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

subgroup

G/H

center

C₂²

D₈×D₁₇

commutator, cosocle

C₂×C₈×C₁₇

C₂³

Frattini

C₂×C₈

C₂²×D₁₇

Fitting

C₁₇×#358

C₂

socle, prime radical

C₂²×C₁₇

C₂×D₈

D₁₇

C₁₇

G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

N	H ≅ G/N	type
77. C₁₇	#2229	semidirect

H ≅ G/N

type

77. C₁₇

#2229

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#11763210	2
Q₁₆×Dic₁₇	2
#11762636	2
#11762635	2
#11762627	2
#11762613	2
C₁₇×#358	2
#2229	17

subgroup

index

Q₁₆×Dic₁₇

C₁₇×#358

Subgroups — #14530950

Conjugacy classes of subgroups

Complements

Maximal subgroups